All About Paradoxes

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Podcast Transcript

2,500 years ago, the Greek philosopher Zeno posited a quandary. If you want wanted to travel from one place to another, you first have to go half the distance, then you have to go half the distance again, and then again. You can do this infinitely and never reach your goal. 

This was one of the first paradoxes known to history. Since then, there have been many many others, which often leave people scratching their heads. 

Learn more about paradoxes, and how to resolve some of them, on this episode of Everything Everywhere Daily.

A paradox is something which is contradictory or something which appears to be contradictory. 

Paradoxes appear in many different areas including logic, mathematics, physics, economics, psychology, and politics. 

Let’s jump right in and talk about the most basic form of paradox, a logical paradox. The most basic would be what’s called the Liar’s Paradox.

The Liar’s Paradox would be the following statement: I always lie.

If in fact I always lie, then that statement is true, and if it is true, then I don’t always lie, which makes the statement false.  The statement is self-contradictory. 

There are many different ways to state the Liar’s Paradox, but in the end, it is pretty much the same. 

Another similar logic paradox was put forward by the British logician Bertrand Russell called the Barber Paradox. It goes like this:

In a town, there is a single barber. This barber shaves all those, and only those, who do not shave themselves. The question then is who shaves the barber?  If the barber shaves himself, then he isn’t the barber. If he doesn’t shave himself, then he has to be shaved by the barber, who is himself. 

This is a self-referential paradox, and most logical paradoxes, even the Liar’s Paradox, are of that type. 

Another great paradox is the surprise test paradox.

Suppose a teacher tells you that at the beginning of class on some day next week, you will get a surprise test.

You figure that the test can’t come in Friday, because if it hasn’t happened through Thursday, you know it will be on Friday and then it won’t be a surprise.

Having eliminated Friday, you realized that it also can’t come on Thursday, because if it hasn’t come on Wednesday, you know it will be Thursday, and then it won’t be a surprise.

You can do this for all the other days of the week and conclude that you won’t get a surprise test.

But then, the teacher gives you the test on Wednesday, which is a surprise. 

You have probably all heard of the question proposed: what would happen if an immovable object collided with an irresistible force? 

This isn’t so much of a paradox as it is a definitional issue. If there is such a thing as an immovable object, then there can’t be such a thing as an irresistible force, and vice versa. 

Definitional paradoxes are easy to create. For example, what would happen if the tallest person in the world met someone taller? Well, then that person wouldn’t be the tallest person in the world.  The statement simply contradicts itself. 

My favorite type of paradox is mathematical paradoxes. They often are true, but they don’t seem like they are true at first glance.

I previously did an entire episode on the Monty Hall Problem which is often called the Monty Hall Paradox. This actually was a huge debate that raged in mathematical circles. 

The paradox is this there are three doors and there is a prize behind one of the doors. If you select one door, and the host shows opens a door with no prize behind it, should you switch doors if given the choice. 

The answer is yes, and I did a previous episode explaining everything. 

Another mathematical paradox is one that you might have encountered before. It states that the number 0.9999999….. repeating is equal to the number 1. 

Many people have a hard time getting their heads around this, but it’s true. The reason it’s true is that it is impossible to come up with a number between .999 repeating and 1. If they were different numbers, then there has to be some number between them. 

Likewise, you can have a definite solution to an infinite sum of numbers. Add up the following infinite list of numbers: 1 + ½ + ¼ + ? + 1/16 etc. 

What does that add up to? You might say that it is impossible to know because there is no last number. However, we do know the answer is 2. The reason why we know it is 2 is the same reason why 0.9999 repeating is equal to 1.  

No matter how close you want to get to 2, the infinite sum can always get closer. 

I bring this up because of the next paradox which I mentioned in the introduction: Zeno’s Paradox.

This is one of the oldest paradoxes known, and it can be stated in several different ways. 

Let’s assume you want to move 2 meters. First, you have to move 1 meter, and then you have to move half a meter, and then a fourth of a meter, etc. According to Zeno, you could never reach your goal because you have to travel an infinite number of distances, each half the length of the previous one. 

The secret to resolving Zeno’s Paradox is that it is mathematically the same as the infinite sum I just mentioned.

However, there is also a physical resolution. In reality, you can’t keep going down in distance forever. There is a minimum distance in the universe called the Planck Length. Once you get down to that distance, you can’t go any further. Likewise, there is a Planck Time, which is the smallest unit of time. 

There are several interesting statistical paradoxes. 

One is called the False Positive or Rare Disease Paradox. 

Suppose you have a rare disease that affects 1 out of every 10,000 people. A test is developed that is 99% accurate.

If you test positive, what are the odds that you have the disease?

You might think that the answer is 99%, but it’s not. 

Let’s say you have a population of 100,000,000 people. In that population, 10,000 people will have the disease. When those 10,000 people are tested, 99% or 9,900 will have a positive test, and 100 people will have a false negative. 

For the rest of the population, 98,990,100 people will return a negative result. However, 1% or 999,900 people will show a false-positive result. 

That means 1,009,800 people will get a positive test results, but for only 9,900, or 1%, will it actually be correct. 

This statistical paradox occurs when the odds of the disease are much less than the accuracy of the test. 

Another interesting statistical paradox is Simpson’s Paradox. Simpson’s Paradox is that trends amongst groups can disappear when the groups are combined. 

For example: in a recent election in Wisconsin there was an issue comparing the educational systems of Wisconsin and Texas. The argument was made that Wisconsin had a better educational system because it had higher average test scores than Texas.

However, when someone looked into it they found someone striking. When they looked at racial subgroups, Texas performed better in every single one. Black students in Texas performed better than in Wisconsin. Latino students in Texas scored better, Asian students scored better, and white students scored better. 

Both of these claims were true. Wisconsin did score better overall, but Texas did score better in every subgroup. How is that possible? It had to do with the size of the groups in each state, and the fact that they had different average scores. 

This can manifest itself in baseball in batting averages, admission rates to college, and other areas. 

There is another paradox called the Will Rodgers Paradox. He first raised it as a joke when he said, “When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states.”

While this was a joke, such a statement can statistically be true. Let’s say you took the smallest person on a basketball team and put them in a group of horse jockeys. Removing the smallest person from the basketball team would increase the average height, and adding them to a group of smaller people would increase their average height as well. 

One of the favorite paradoxes of movies and science fiction is time travel paradoxes. The most well-known of which is the Grandfather Paradox.

Let’s assume you go back in time and kill your grandfather when he was a child. You would never be born, when means you wouldn’t be around to go back in time to kill your grandfather. 

There are all sorts of time travel paradoxes, which have been used as plot devices in many movies. 

There are several things called paradoxes in the physical world which really aren’t paradoxes per se. 

One is the potato paradox. Let’s say a farmer has 100 pounds of potatoes and the potatoes are 99% water and 1% solids. 

The farmer leaves them in the sun until the water level goes down to only 98% water. When he weighs that 98% water potatoes, it now weighs 50 pounds, having lost half the weight.  How is that possible?

In the initial weight, 1% of the 100 pounds was solid, or 1 pound. The weight of the solids doesn’t change after the potatoes dry. The new ratio is 2 solid to 98 water, or alternatively 1 solid to 49 water. If the solid remains 1 pound, the water has to weigh 49 pounds, for a total of 50 pounds. 

In nutrition, there was something called the French and Israeli paradoxes. Researches in the 60s thought that saturated fat caused heart disease, but France consumed high levels of saturated fat and had low heart disease. Likewise, Israel had low levels of saturated fat consumption and high heart disease. 

The answer of course is that there is no paradox in such cases. The assumptions are usually just wrong, and that is what has been discovered since then. 

Voting has many paradoxes. Arrow’s Paradox was proven by Nobel Prize winning economist Kenneth Arrow. He showed that no ranked voting system with three or more choices can always accurately reflect voters’ desires. This is because voting, unlike mathematics, isn’t transitive. 

Fenno’s Paradox states that most polls show that American’s have low approval for Congress as a body, yet most individual members of congress have a very high rate of reelection. 

The apportionment of representatives has several paradoxes. One is called the Alabama Paradox. It was discovered in 1880 that increasing the size of the House of Representative would actually have decreased the number of representatives in Alabama by 1.

Likewise, there is the New State Paradox. When Oklahoma was added as a state, New York lost a seat and Maine gained one, even though the total size of the house increased. 

There are dozens and dozens of paradoxes, both literal and figurative. Sometimes they are a true paradox, and other times they are just a puzzle to be figured out for it to make sense. 

The associate producers of Everything Everywhere Daily are Thor Thomsen and Peter Bennett. 

Today’s review comes from listener guisa1981 on Apple Podcasts in Canada. They write:

A new favorite

Original topics that are superbly summarized and delivered. I’ve been binge-listening this podcast for a couple of days. Well done sir!

Thank you, guisa1981. It is always a comfort to know that you are bingeworth.

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